Increasing and Decreasing IntervalsWhere Does the Fun End?

Anne DudleyMichael Holtfrerich

Joshua WhitneyGlendale Community College

Glendale, Arizona

Determine the largest interval on which each function (on the handout) is increasing.

Once you have determined your answers, talk to neighbors about their answers.

Increasing & DecreasingActivity 1

1. (–1, 1]2. (–1, 1)3. [–1, 1]4. 5. Other

Graph 1 IncreasingUse your clicker to indicate your answer.

What is the largest interval on which the function is increasing?

1 2 3 4 5

0%

50%

0%

50%

0%

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

x

y

f(x) = -(1/3) x3 + x ,11,

1. 2. 3. 4. Other

Graph 2 IncreasingUse your clicker to indicate your answer.

What is the largest interval on which the function is increasing?

1 2 3 4

0% 0%

100%

0%

-1 1 2 3

1

2

3

4

5

x

y

1. 2. 3. 4. Other

Graph 2 DecreasingUse your clicker to indicate your answer.

What is the largest interval on which the function is decreasing?

1 2 3 4

25% 25%25%25%

-1 1 2 3

1

2

3

4

5

x

y

If you used closed intervals for your two previous answers, are you OK with the graph being both increasing and decreasing at x = 1?

Will it confuse your students?

Discussion

Graph 3 IncreasingUse your clicker to indicate your answer.

What is the largest interval on which the function is increasing?

1 2 3 4

25% 25%25%25%

-1 1

1

2

3

4

5

x

y

1. 2. 3. 4. Other

Graph 4 IncreasingUse your clicker to indicate your answer.

Where is the function increasing?

1 2 3 4

50% 50%

0%0%

1. 2. 3. 4. Other

-2 -1 1 2

-3

-2

-1

1

2

3

4

5

6

7

x

y

f(x) = x3+3

Ostebee & Zorn Calculus, 2nd Edition (p.56)Definition: Let I denote the interval (a, b). A function f is increasing on I if

whenever a < x1 < x2 < b. f is decreasing on I if whenever a < x1 < x2 < b.

Fact: If for all x in I, then f increases on I. If for all x in I, then f decreases on I.

What Do Textbooks Say?

Graph 1 Increasing (Ostebee & Zorn )Use your clicker to indicate your answer.

What is the largest interval on which the function is increasing?

1 2 3 4

0% 0%0%

100%

1. 2. 3. 4. Other

1,1 1,1 1,1

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

x

y

f(x) = -(1/3) x3 + x

Hughes-Hallett Calculus, 5th edition (p. 165) and Swokowski Calculus, 2nd edition (p.147)Let f(x) be continuous on [a, b], and differentiable on (a, b).If f’(x) > 0 on a < x < b, then f is increasing on .

If f’(x) ≥ 0 on a < x < b, then f is non-decreasing on .

What Do Textbooks Say?

Graph 1 Increasing (Hughes-Hallett)Use your clicker to indicate your answer.

What is the largest interval on which the function is increasing?

1 2 3 4

0% 0%

100%

0%

1. 2. 3. 4. Other

1,1 1,1 1,1

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

x

y

f(x) = -(1/3) x3 + x

Larson Edwards Calculus, 5th edition (p. 219) A function f is increasing on an interval if for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).A function f is decreasing on an interval if for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2).

What Do Textbooks Say?

The Problem with “Any”

The following function isnot increasing on [a,b].

But it does fit the precedingdefinition.

For the two numbers x1< x2 , f(x1) < f(x2) should imply that f(x) is increasing on [a,b].

x

y

a bx1 x2

Larson Edwards Calculus, 5th edition (p. 219) (Theorem)

Let f (x) be continuous on [a, b], and differentiable on (a, b).

If f’(x) > 0 for all x in (a, b), then f is increasing on [a, b].

If f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b].

If f’(x) = 0 for all x in (a, b), then f is constant on [a, b].

All textbook examples and the answers in homework are open intervals.

What Do Textbooks Say?

Cynthia Young Precalculus, 1st edition (p. 128)A function f is increasing on an open interval I if for any x1 and x2 in I, where x1 < x2, then f(x1) < f(x2).

A function f is decreasing on an open interval I if for any x1 and x2 in I, where x1 < x2, then f(x1) > f(x2).

What Do Textbooks Say?

Two Point Idea of Increasing

Slope of the Tangent Line Idea of Increasing(one point idea)

Two Differing Ideas

-2 -1 1 2

-3

-2

-1

1

2

3

4

5

6

7

x

y

f(x) = x3+3

Now determine the largest interval on which the function is increasing for the three new examples.

Be prepared to clicker your choice.

Apply the Definitions

Graph 5 IncreasingUse your clicker to indicate your answer.

What is the largest interval on which the function is increasing?

1 2 3 4

50% 50%

0%0%

1. 2. 3. 4. Other

1 2 3

-1

1

2

3

x

y

1. 2. 3. 4.

Graph 6 IncreasingUse your clicker to indicate your answer.

What is the largest interval on which the function is increasing?

1 2 3 4

0% 0%

100%

0%

1 2 3

1

2

3

4

5

x

y

Graph 7 (AKA Final Exam) Use your clicker to indicate your answer.

Where is the function increasing?

1 2 3 4 5

20% 20% 20%20%20%

-1 1 2 3 4 5 6

-1

1

2

3

4

5

6

x

y1. 2. 3. 4. 5. Other

We propose this definition for all textbooks at the College Algebra (Pre-Calculus) level and below:A function f is increasing on an open interval I if for all x1 and x2 in I, where x1 < x2, then f(x1) < f(x2).A function f is decreasing on an open interval I if for all x1 and x2 in I, where x1 < x2, then f(x1) > f(x2).

Dudfreney Intervals

Historical Ideas

Introduction to Infinitesimal Analysis,O. Veblen, 1907

Historical Ideas

Theory of Functionsof Real Variables,J. Pierpont, 1905

Historical Ideas

Differential and Integral Calculus,G. Osborne, 1907

Historical Ideas

An Elementary Treatise on the Calculus,G. Gibson, 1901

Anne – Anne.Dudley@gccaz.edu

Michael – M.Holtfrerich@gccaz.edu

Josh – Joshua.Whitney@gccaz.edu

Thanks for Your Participation